RV coefficient

<h2 id="definitions">Definitions</h2>
The definition of the RV-coefficient makes use of ideas<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued <a href="/facts/Random_variables/TwTBXnLT">random variables</a>. Note that standard usage is to have matrices for the variances and covariances of vector random variables. 
Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.
Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by

Σ
          
            X
            Y
          
        
        =
        E
        ⁡
        (
        X
        
          Y
          
            ⊤
          
        
        )
        
        ,
      
    
    {\displaystyle \Sigma _{XY}=\operatorname {E} (XY^{\top })\,,}

then the scalar-valued covariance (denoted by COVV) is defined by<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

COVV
        ⁡
        (
        X
        ,
        Y
        )
        =
        Tr
        ⁡
        (
        
          Σ
          
            X
            Y
          
        
        
          Σ
          
            Y
            X
          
        
        )
        
        .
      
    
    {\displaystyle \operatorname {COVV} (X,Y)=\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})\,.}

The scalar-valued variance is defined correspondingly:

VAV
        ⁡
        (
        X
        )
        =
        Tr
        ⁡
        (
        
          Σ
          
            X
            X
          
          
            2
          
        
        )
        
        .
      
    
    {\displaystyle \operatorname {VAV} (X)=\operatorname {Tr} (\Sigma _{XX}^{2})\,.}

With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Then the RV-coefficient is defined by<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

R
          V
        
        (
        X
        ,
        Y
        )
        =
        
          
            
              COVV
              ⁡
              (
              X
              ,
              Y
              )
            
            
              VAV
              ⁡
              (
              X
              )
              VAV
              ⁡
              (
              Y
              )
            
          
        
        
        .
      
    
    {\displaystyle \mathrm {RV} (X,Y)={\frac {\operatorname {COVV} (X,Y)}{\sqrt {\operatorname {VAV} (X)\operatorname {VAV} (Y)}}}\,.}

<h2 id="shortcoming-of-the-coefficient-and-adjusted-version">Shortcoming of the coefficient and adjusted version</h2>
Even though the coefficient takes values between 0 and 1 by construction, it seldom attains values close to 1 as the denominator is often too large with respect to the maximal attainable value of the denominator.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 
Given known diagonal blocks 
 
 
 
 
 Σ
 
 X
 X
 
 
 
 
 {\displaystyle \Sigma _{XX}}
 
 and 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 
 
 {\displaystyle \Sigma _{YY}}
 
 of dimensions 
 
 
 
 p
 ×
 p
 
 
 {\displaystyle p\times p}
 
 and 
 
 
 
 q
 ×
 q
 
 
 {\displaystyle q\times q}
 
 respectively, assuming that 
 
 
 
 p
 ≤
 q
 
 
 {\displaystyle p\leq q}
 
 without loss of generality, it has been proved<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> that the maximal attainable numerator is 
 
 
 
 Tr
 ⁡
 (
 
 Λ
 
 X
 
 
 Π
 
 Λ
 
 Y
 
 
 )
 ,
 
 
 {\displaystyle \operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y}),}
 
 
where 
 
 
 
 
 Λ
 
 X
 
 
 
 
 {\displaystyle \Lambda _{X}}
 
 (resp. 
 
 
 
 
 Λ
 
 Y
 
 
 
 
 {\displaystyle \Lambda _{Y}}
 
) denotes the diagonal matrix of the eigenvalues of 
 
 
 
 
 Σ
 
 X
 X
 
 
 
 
 {\displaystyle \Sigma _{XX}}
 
(resp. 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 
 
 {\displaystyle \Sigma _{YY}}
 
) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and 
 
 
 
 Π
 
 
 {\displaystyle \Pi }
 
 is the 
 
 
 
 p
 ×
 q
 
 
 {\displaystyle p\times q}
 
 matrix 
 
 
 
 (
 
 I
 
 p
 
 
  
 
 0
 
 p
 ×
 (
 q
 −
 p
 )
 
 
 )
 
 
 {\displaystyle (I_{p}\ 0_{p\times (q-p)})}
 
.
In light of this, Mordant and Segers<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads

RV
              ¯
            
          
        
        (
        X
        ,
        Y
        )
        =
        
          
            
              Tr
              ⁡
              (
              
                Σ
                
                  X
                  Y
                
              
              
                Σ
                
                  Y
                  X
                
              
              )
            
            
              Tr
              ⁡
              (
              
                Λ
                
                  X
                
              
              Π
              
                Λ
                
                  Y
                
              
              )
            
          
        
        =
        
          
            
              Tr
              ⁡
              (
              
                Σ
                
                  X
                  Y
                
              
              
                Σ
                
                  Y
                  X
                
              
              )
            
            
              
                ∑
                
                  j
                  =
                  1
                
                
                  m
                  i
                  n
                  (
                  p
                  ,
                  q
                  )
                
              
              (
              
                Λ
                
                  X
                
              
              
                )
                
                  j
                  ,
                  j
                
              
              (
              
                Λ
                
                  Y
                
              
              
                )
                
                  j
                  ,
                  j
                
              
            
          
        
        .
      
    
    {\displaystyle {\bar {\operatorname {RV} }}(X,Y)={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y})}}={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\sum _{j=1}^{min(p,q)}(\Lambda _{X})_{j,j}(\Lambda _{Y})_{j,j}}}.}

The impact of this adjustment is clearly visible in practice.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Congruence_coefficient/8F8zPx8b">Congruence coefficient</a></li>
<li><a href="/facts/Distance_correlation/8YMXWXP8">Distance correlation</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Robert, P.; Escoufier, Y. (1976). "A Unifying Tool for Linear Multivariate Statistical Methods: The RV-Coefficient". Applied Statistics. 25 (3): 257–265. doi:10.2307/2347233. JSTOR 2347233. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Abdi, Hervé (2007). Salkind, Neil J (ed.). RV coefficient and congruence coefficient. Thousand Oaks. ISBN 978-1-4129-1611-0. <a href="978-1-4129-1611-0" target="_blank">978-1-4129-1611-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Robert, P.; Escoufier, Y. (1976). "A Unifying Tool for Linear Multivariate Statistical Methods: The RV-Coefficient". Applied Statistics. 25 (3): 257–265. doi:10.2307/2347233. JSTOR 2347233. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Ferath Kherif; Jean-Baptiste Poline; Sébastien Mériaux; Habib Banali; Guillaume Plandin; Matthew Brett (2003). "Group analysis in functional neuroimaging: selecting subjects using similarity measures" (PDF). NeuroImage. 20 (4): 2197–2208. doi:10.1016/j.neuroimage.2003.08.018. PMID 14683722. <a href="https://hal-cea.archives-ouvertes.fr/cea-00371054/file/Kherifetal_NeuroImage.pdf" target="_blank">https://hal-cea.archives-ouvertes.fr/cea-00371054/file/Kherifetal_NeuroImage.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Herve Abdi; Joseph P. Dunlop; Lynne J. Williams (2009). "How to compute reliability estimates and display confidence and tolerance intervals for pattern classiffers using the Bootstrap and 3-way multidimensional scaling (DISTATIS)". NeuroImage. 45 (1): 89–95. doi:10.1016/j.neuroimage.2008.11.008. PMID 19084072. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Escoufier, Y. (1973). "Le Traitement des Variables Vectorielles". Biometrics. 29 (4). International Biometric Society: 751–760. doi:10.2307/2529140. JSTOR 2529140. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Escoufier, Y. (1973). "Le Traitement des Variables Vectorielles". Biometrics. 29 (4). International Biometric Society: 751–760. doi:10.2307/2529140. JSTOR 2529140. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Escoufier, Y. (1973). "Le Traitement des Variables Vectorielles". Biometrics. 29 (4). International Biometric Society: 751–760. doi:10.2307/2529140. JSTOR 2529140. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Escoufier, Y. (1973). "Le Traitement des Variables Vectorielles". Biometrics. 29 (4). International Biometric Society: 751–760. doi:10.2307/2529140. JSTOR 2529140. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Pucetti, G. (2019). "Measuring Linear Correlation Between Random Vectors". SSRN. <a href="https://dx.doi.org/10.2139/ssrn.3116066s" target="_blank">https://dx.doi.org/10.2139/ssrn.3116066s</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Mordant Gilles; Segers Johan (2022). "Measuring dependence between random vectors via optimal transport,". Journal of Multivariate Analysis. 189. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Mordant Gilles; Segers Johan (2022). "Measuring dependence between random vectors via optimal transport,". Journal of Multivariate Analysis. 189. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Mordant Gilles; Segers Johan (2022). "Measuring dependence between random vectors via optimal transport,". Journal of Multivariate Analysis. 189. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

RV coefficient open-in-new

RV coefficient